Like any complex manifold, \(\mathcal {T}_g\)  can be equipped with the Kobayashi pseudo-metric, see [1]} for a comprehensive introduction, see also page 167 of [2]}. If the Kobayashi pseudo-metric is complete (or non-degenerate), see page 168 of [1]}, then the complex manifold is called Kobayashi hyperbolic, or a hyperbolic complex space.
More classically, a metric on \(\mathcal {T}_g\)  is introduced by using (the existence and uniqueness of) minimal dilation quasi-conformal maps between compact Riemann surfaces equipped with holomorphic quadratic differentials of the same typei.e, the number and the multiplicity of the zeroes of the quadratic differentials match, this metric is called the Teichmüller metric, see section 5.1.3 of [2]} for more details. H. L Royden showed that the Kobayashi pseudo-metric is equivalent to the Teichmüller metric, see [5]} or theorem 6.21 in [2]}, and since the Teichmüller metric is complete, see theorem 5.4 in [2]}, it follows that \(\mathcal {T}_g\)  is Kobayashi hyperbolic.
